![]() |
Mathbox for Alan Sare |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > sspwimpALT2 | Structured version Visualization version GIF version |
Description: If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
sspwimpALT2 | ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3476 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | elpwi 4608 | . . . . 5 ⊢ (𝑥 ∈ 𝒫 𝐴 → 𝑥 ⊆ 𝐴) | |
3 | id 22 | . . . . 5 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵) | |
4 | 2, 3 | sylan9ssr 3995 | . . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ⊆ 𝐵) |
5 | elpwg 4604 | . . . . 5 ⊢ (𝑥 ∈ V → (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)) | |
6 | 5 | biimpar 476 | . . . 4 ⊢ ((𝑥 ∈ V ∧ 𝑥 ⊆ 𝐵) → 𝑥 ∈ 𝒫 𝐵) |
7 | 1, 4, 6 | sylancr 585 | . . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 𝐵) |
8 | 7 | ex 411 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵)) |
9 | 8 | ssrdv 3987 | 1 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2104 Vcvv 3472 ⊆ wss 3947 𝒫 cpw 4601 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-v 3474 df-in 3954 df-ss 3964 df-pw 4603 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |